8. Snow. Based on meteorological data for the past

1. t-models, part I. Using the t tables, software, or a calculator, estimate a) the critical value of t for a 90% confidence interval with b) the critical value of t for a 98% confidence interval with c) the P-value for with 4 degrees of freedom. d) the P-value for with 22 degrees of freedom.

2. t-models, part II. Using the t tables, software, or a calculator, estimate a) the critical value of t for a 95% confidence interval with b) the critical value of t for a 99% confidence interval with c) the P-value for with 41 degrees of freedom. d) the P-value for with 12 degrees of freedom.

3. t-models, part III. Describe how the shape, center, and spread of t-models change as the number of degrees of freedom increases.

4. t-models, part IV (last one!). Describe how the critical value of t for a 95% confidence interval changes as the number of degrees of freedom increases.

5. Cattle. Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of pounds. Some students wrote the following conclusions. Did anyone interpret the interval correctly? Explain any misinterpretations. a) 95% of the cows studied gained between 45 and 67 pounds. b) Were 95% sure that a cow fed this supplement will gain between 45 and 67 pounds. c) Were 95% sure that the average weight gain among the cows in this study was between 45 and 67 pounds. d) The average weight gain of cows fed this supplement will be between 45 and 67 pounds 95% of the time. e) If this supplement is tested on another sample of cows, there is a 95% chance that their average weight gain will be between 45 and 67 pounds.

6. Teachers. Software analysis of the salaries of a random sample of 288 Nevada teachers produced the confidence interval shown below. Which conclusion is correct? Whats wrong with the others? t -Inter val for _ : with 90.00% Confidence, 38944 < _ (TchPay) < 42893 a) If we took many random samples of 288 Nevada teachers, about 9 out of 10 of them would produce this confidence interval. b) If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce a confidence interval that contained the mean salary of all Nevada teachers. c) About 9 out of 10 Nevada teachers earn between $38,944 and $42,893. d) About 9 out of 10 of the teachers surveyed earn between $38,944 and $42,893. e) We are 90% confident that the average teacher salary in the United States is between $38,944 and $42,893.

7. Meal plan. After surveying students at Dartmouth College, a campus organization calculated that a 95% confidence interval for the mean cost of food for one term (of three in the Dartmouth trimester calendar) is ($1102, $1290). Now the organization is trying to write its report and is considering the following interpretations. Comment on each. a) 95% of all students pay between $1102 and $1290 for food. b) 95% of the sampled students paid between $1102 and $1290. c) Were 95% sure that students in this sample averaged between $1102 and $1290 for food. d) 95% of all samples of students will have average food costs between $1102 and $1290. e) Were 95% sure that the average amount all students pay is between $1102 and $1290.

8. Snow. Based on meteorological data for the past century, a local TV weather forecaster estimates that the regions average winter snowfall is with a margin of error of inches. Assuming he used a 95% confidence interval, how should viewers interpret this news? Comment on each of these statements: a) During 95 of the last 100 winters, the region got between and of nows . b) Theres a 95% chance the region will get between and of snow this winter. c) There will be between and of snow on the ground for 95% of the winter days. d) Residents can be 95% sure that the areas average snowfall is between and . e) Residents can be 95% confident that the average snowfall during the last century was between and per winter.

9. Pulse rates. A medical researcher measured the pulse rates (beats per minute) of a sample of randomly selected adults and found the following Students t-based confidence interval: a) Explain carefully what the software output means. b) Whats the margin of error for this interval? c) If the researcher had calculated a 99% confidence interval, would the margin of error be larger or smaller? Explain.

10. Crawling. Data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl: a) Explain carefully what the software output means. b) What is the margin of error for this interval? c) If the researcher had calculated a 90% confidence interval, would the margin of error be larger or smaller? Explain.

11. CEO compensation. A sample of 20 CEOs from the Forbes 500 shows total annual compensations ranging from a minimum of $0.1 to $62.24 million. The average for these 20 CEOs is $7.946 million. Heres a histogram: Based on these data, a computer program found that a 95% confidence interval for the mean annual compensation of all Forbes 500 CEOs is (1.69, 14.20) $ million. Why should you be hesitant to trust this confidence interval?

12. Credit card charges. A credit card company takes a random sample of 100 cardholders to see how much they charged on their card last month. Heres a histogram. A computer program found that the resulting 95% confidence interval for the mean amount spent in March 2005 is ( Explain why the analysts didnt find the confidence interval useful, and explain

13. Normal temperature. The researcher described in Exercise 9 also measured the body temperatures of that randomly selected group of adults. Here are summaries of the data he collected. We wish to estimate the average (or normal) temperature among the adult population. a) Check the conditions for creating a t-interval. b) Find a 98% confidence interval for mean body temperature. c) Explain the meaning of that interval. d) Explain what 98% confidence means in this context. e) is commonly assumed to be normal. Do these data suggest otherwise? Explain.

14. Parking. Hoping to lure more shoppers downtown, a city builds a new public parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period (44 weekdays), daily fees collected averaged $126, with a standard deviation of $15. a) What assumptions must you make in order to use these statistics for inference? b) Write a 90% confidence interval for the mean daily income this parking garage will generate. c) Interpret this confidence interval in context. d) Explain what 90% confidence means in this context. e) The consultant who advised the city on this project predicted that parking revenues would average $130 per day. Based on your confidence interval, do you think the consultant was correct? Why?

15. Normal temperatures, part II. Consider again the statistics about human body temperature in Exercise 13. a) Would a 90% confidence interval be wider or narrower than the 98% confidence interval you calculated before? Explain. (Dont compute the new interval.) b) What are the advantages and disadvantages of the 98% confidence interval? c) If we conduct further research, this time using a sample of 500 adults, how would you expect the 98% confidence interval to change? Explain. d) How large a sample might allow you to estimate the mean body temperature to within 0.1 degrees with 98% confidence?

16. Parking II. Suppose that, for budget planning purposes, the city in Exercise 14 needs a better estimate of the mean daily income from parking fees. a) Someone suggests that the city use its data to create a 95% confidence interval instead of the 90% interval first created. How would this interval be better for the city? (You need not actually create the new interval.) b) How would the 95% interval be worse for the planners? c) How could they achieve an interval estimate that would better serve their planning needs? d) How many days worth of data should they collect to have 95% confidence of estimating the true mean to within $3?

17. Speed of light. In 1882 Michelson measured the speed of light (usually denoted c as in Einsteins famous equation ). His values are in km/sec and have 299,000 subtracted from them. He reported the results of 23 trials with a mean of 756.22 and a standard deviation of 107.12. a) Find a 95% confidence interval for the true speed of light from these statistics. b) State in words what this interval means. Keep in mind that the speed of light is a physical constant that, as far as we know, has a value that is true throughout the universe. c) What assumptions must you make in order to use your method?

18. Better light. After his first attempt to determine the speed of light (described in Exercise 17), Michelson conducted an improved experiment. In 1897 he reported results of 100 trials with a mean of 852.4 and a standard deviation of 79.0. a) What is the standard error of the mean for these data? b) Without computing it, how would you expect a 95% confidence interval for the second experiment to differ from the confidence interval for the first? Note at least three specific reasons why they might differ, and indicate the ways in which these differences would change the interval. c) According to Stigler (who reports these values), the true speed of light is 299,710.5 km/sec, corresponding to a value of 710.5 for Michelsons 1897 measurements. What does this indicate about Michelsons two experiments? Explain, using your confidence interval.

19. Departures. What are the chances your flight will leave on time? The U.S. Bureau of Transportation Statistics of the Department of Transportation publishes information about airline performance. Here are a histogram and summary statistics for the percentage of flights departing on time each month from 1995 thru 2006. There is no evidence of a trend over time. (The correlation of On Time Departure% with time is ) a) Check the assumptions and conditions for inference. b) Find a 90% confidence interval for the true percentage of flights that depart on time. c) Interpret this interval for a traveler planning to fly.

20. Late arrivals. Will your flight get you to your destination on time? The U.S. Bureau of Transportation Statistics reported the percentage of flights that were late each month from 1995 through 2006. Heres a histogram, along with some summary statistics: We can consider these data to be a representative sample of all months. There is no evidence of a time trend a) Check the assumptions and conditions for inference about the mean. b) Find a 99% confidence interval for the true percentage of flights that arrive late. c) Interpret this interval for a traveler planning to fly.

21. For Example, 2nd look. This chapters For Examples looked at mirex contamination in farmed salmon. We first found a 95% confidence interval for the mean concentration to be 0.0834 to 0.0992 parts per million. Later we rejected the null hypothesis that the mean did not exceed the EPAs recommended safe level of 0.08 ppm based on a P-value of 0.0027. Explain how these two results are consistent. Your explanation should discuss the confidence level, the P-value, and the decision.

22. Hot Dogs. A nutrition lab tested 40 hot dogs to see if their mean sodium content was less than the 315 mg upper limit set by regulations for "reduced sodium" franks. The lab failed to reject the hypothesis that the hot dogs did not meet this requirement, with a P-value of 0.141. A 90% confidence interval estimated the mean sodium content for this kind of hot dog at 317.1 to 316.8 mg. Explain how these two results are consistent. Your explanation should discuss the confidence level, the P-value, and the decision.

23. Pizza. A researcher tests whether the mean cholesterol level among those who eat frozen pizza exceeds the value considered to indicate a health risk. She gets a P-value of 0.07. Explain in this context what the "7%" represents.

24. Golf balls. The United States Golf Association (USGA) sets performance standards for golf balls. For example, the initial velocity of the ball may not exceed 150 feet per second when measured by an apparatus approved by the USGA. Suppose a manufacturer introduces a new kind of ball and provides a sample for testing. Based on the mean speed in the test, the USGA comes up with a P-value of 0.34. Explain in this context what the "34%" represents.

25. TV safety. The manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. Since some larger sets weigh nearly 300 pounds, the companys safety inspectors have set a standard of ensuring that the stands can support an average of over 500 pounds. Their inspectors regularly subject a random sample of the stands to increasing weight until they fail. They test the hypothesis against , using the level of significance . If the sample of stands fail to pass this safety test, the inspectors will not certify the product for sale to the general public. a) Is this an upper-tail or lower-tail test? In the context of the problem, why do you think this is important? b) Explain what will happen if the inspectors commit a Type I error. c) Explain what will happen if the inspectors commit a Type II error.

26. Catheters. During an angiogram, heart problems can be examined via a small tube (a catheter) threaded into the heart from a vein in the patients leg. Its important that the company that manufactures the catheter maintain a diameter of 2.00 mm. (The standard deviation is quite small.) Each day, quality control personnel make several measurements to test against at a significance level of . If they discover a problem, they will stop the manufacturing process until it is corrected. a) Is this a one-sided or two-sided test? In the context of the problem, why do you think this is important? b) Explain in this context what happens if the quality control people commit a Type I error. c) Explain in this context what happens if the quality control people commit a Type II error.

27. TV safety revisited. The manufacturer of the metal TV stands in Exercise 25 is thinking of revising its safety test. a) If the companys lawyers are worried about being sued for selling an unsafe product, should they increase or decrease the value of ? Explain. b) In this context, what is meant by the power of the test? c) If the company wants to increase the power of the test, what options does it have? Explain the advantages and disadvantages of each option.

28. Catheters again. The catheter company in Exercise 26 is reviewing its testing procedure. a) Suppose the significance level is changed to . Will the probability of a Type II error increase, decrease, or remain the same? b) What is meant by the power of the test the company conducts? c) Suppose the manufacturing process is slipping out of proper adjustment. As the actual mean diameter of the catheters produced gets farther and farther above the desired 2.00 mm, will the power of the quality control test increase, decrease, or remain the same? d) What could they do to improve the power of the test?

29. Marriage. In 1960, census results indicated that the age at which American men first married had a mean of 23.3 years.It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 40 years. a) Write appropriate hypotheses. b) We plan to test our hypothesis by selecting a random sample of 40 men who married for the first time last year. Do you think the necessary assumptions for inference are satisfied? Explain. c) Describe the approximate sampling distribution model for the mean age in such samples. d) The men in our sample married at an average age of 24.2 years, with a standard deviation of 5.3 years. Whats the P-value for this result? e) Explain (in context) what this P-value means. f) Whats your conclusion?

30. Fuel economy. A company with a large fleet of cars hopes to keep gasoline costs down and sets a goal of attaining a fleet average of at least 26 miles per gallon. To see if the goal is being met, they check the gasoline usage for 50 company trips chosen at random, finding a mean of 25.02 mpg and a standard deviation of 4.83 mpg. Is this strong evidence that they have failed to attain their fuel economy goal? a) Write appropriate hypotheses. b) Are the necessary assumptions to make inferences satisfied? c) Describe the sampling distribution model of mean fuel economy for samples like this. d) Find the P-value. e) Explain what the P-value means in this context. f) State an appropriate conclusion.

31. Ruffles. Students investigating the packaging of potato chips purchased 6 bags of Lays Ruffles marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.3, 28.2, 29.1, 28.7, 28.9, 28.5. a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a 95% confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the companys stated net weight of 28.3 grams.

32. Doritos. Some students checked 6 bags of Doritos marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.2, 28.5, 28.7, 28.9, 29.1, 29.5. a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a 95% confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the companys stated net weight of 28.3 grams.

33. Popcorn. Yvon Hopps ran an experiment to test optimum power and time settings for microwave popcorn. His goal was to find a combination of power and time that would deliver high-quality popcorn with less than 10% of the kernels left unpopped, on average. After experimenting with several bags, he determined that power 9 at 4 minutes was the best combination. a) He concluded that this popping method achieved the 10% goal. If it really does not work that well, what kind of error did Hopps make? b) To be sure that the method was successful, he popped 8 more bags of popcorn (selected at random) at this setting. All were of high quality, with the following percentages of uncooked popcorn: 7, 13.2, 10, 6, 7.8, 2.8, 2.2, 5.2. Does this provide evidence that he met his goal of an average of no more than 10% uncooked kernels? Explain.

34. Ski wax. Bjork Larsen was trying to decide whether to use a new racing wax for cross-country skis. He decided that the wax would be worth the price if he could average less than 55 seconds on a course he knew well, so he planned to test the wax by racing on the course 8 times. a) Suppose that he eventually decides not to buy the wax, but it really would lower his average time to below 55 seconds. What kind of error would he have made? b) His 8 race times were 56.3, 65.9, 50.5, 52.4, 46.5, 57.8, 52.2, and 43.2 seconds. Should he buy the wax? Explain.

Chips Ahoy. In 1998, as an advertising campaign, the Nabisco Company announced a 1000 Chips Challenge, claiming that every 18-ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, 12, no. 1[1999]) a) Check the assumptions and conditions for inference. Comment on any concerns you have. b) Create a 95% confidence interval for the average number of chips in bags of Chips Ahoy cookies. c) What does this evidence say about Nabiscos claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

36. Yogurt. Consumer Reports tested 14 brands of vanilla yogurt and found these numbers of calories per serving: a) Check the assumptions and conditions for inference. b) Create a 95% confidence interval for the average calorie content of vanilla yogurt. c) A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

37. Maze. Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take rats an average of about one minute to solve. He tests one maze on several rats, collecting the data shown. a) Plot the data. Do you think the conditions for inference are satisfied? Explain. b) Test the hypothesis that the mean completion time for this maze is 60 seconds. What is your conclusion? c) Eliminate the outlier, and test the hypothesis again. What is your conclusion? d) Do you think this maze meets the one-minute average requirement? Explain.

38. Braking. A tire manufacturer is considering a newly designed tread pattern for its all-weather tires. Tests have indicated that these tires will provide better gas mileage and longer tread life. The last remaining test is for braking effectiveness. The company hopes the tire will allow a car traveling at 60 mph to come to a complete stop within an average of 125 feet after the brakes are applied. They will adopt the new tread pattern unless there is strong evidence that the tires do not meet this objective. The distances (in feet) for 10 stops on a test track were 129, 128, 130, 132, 135, 123, 102, 125, 128, and 130. Should the company adopt the new tread pattern? Test an appropriate hypothesis and state your conclusion. Explain how you dealt with the outlier and why you made the recommendation you did.

39. Driving distance. How far do professional golfers drive a ball? (For non-golfers, the drive is the shot hit from a tee at the start of a hole and is typically the longest shot.) Heres a histogram of the average driving distances of the 202 leading professional golfers in 2006 along with summary statistics. a) Find a 95% confidence interval for the mean drive distance. b) Interpreting this interval raises some problems. Discuss. c) The data are the mean driving distance for each golfer. Is that a concern in interpreting the interval? (Hint: Review the What Can Go Wrong warnings of Chapter 9. Chapter 9?! Yes, Chapter 9.)

40. Wind power. Should you generate electricity with your own personal wind turbine? That depends on whether you have enough wind on your site. To produce enough energy, your site should have an annual average wind speed above 8 miles per hour, according to the Wind Energy Association. One candidate site was monitored for a year, with wind speeds recorded every 6 hours. A total of 1114 readings of wind speed averaged 8.019 mph with a standard deviation of 3.813 mph. Youve been asked to make a statistical report to help the landowner decide whether to place a wind turbine at this site. a) Discuss the assumptions and conditions for using Students t inference methods with these data. Here are some plots that may help you decide whether the methods can be used: b) What would you tell the landowner about whether this site is suitable for a small wind turbine? Explain.